To multiply \(9\sqrt{7} \cdot 6\sqrt{7}\), you can follow these steps:
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Multiply the coefficients (the numbers outside the square roots): \[ 9 \cdot 6 = 54. \]
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Multiply the square roots: \[ \sqrt{7} \cdot \sqrt{7} = \sqrt{7^2} = 7. \]
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Combine the results: \[ 54 \cdot 7 = 378. \]
Thus, the exact answer is: \[ \boxed{378}. \]